Problem: A regular polygon has an exterior angle that measures $15$ degrees. How many sides does the polygon have?
Explanation: The sum of the exterior angles of a polygon is $360^\circ$ as long as we take only one exterior angle per vertex. The polygon is regular, so all of the exterior angles have the same degree measure of $15$ degrees. If the polygon has $n$ sides, then the sum of the exterior angles is $15n=360$. So $n=24$ and the polygon has $\boxed{24}$ sides.